This concept is made quantitative in Figure REF . The figure displays the center of mass energy \({\sqrt{s\,}}_{\hspace{-2.0pt}p}\)  that a proton collider must possess to be “equivalent” to a muon collider of a given energy \(E_{\rm {cm}}=\sqrt{s\,}_{\hspace{-2.0pt}\mu }\) . Equivalence is defined [1]}, [2]}, [3]} in terms of the pair production cross-section for heavy particles, with mass close to the muon collider kinematical threshold of \(\sqrt{s\,}_{\hspace{-2.0pt}\mu }/2\) . The equivalent \({\sqrt{s\,}}_{\hspace{-2.0pt}p}\)  is the proton collider center of mass energy for which the cross-sections at the two colliders are equal.
